There are many ways of doing this, but on the NG I just can't be bothered to tune my way down to touchdown.
The aircraft monitors position and will report deviations. GPS, IRS, DME/DME, etc, etc, and I have never had anything resembling map shift.
The displays are left on MAP, raw data ILS is available on the HSI. This gives excellent awareness of my position during the approach.
The Kazan guys were on the ILS , but seemed to have some problems getting there.